1. Introduction

(a) double auction: any procedure in which buyers and sellers interact

to arrange trade.

i. contrast with the passive role of the seller in most auction models

ii. can be dynamic or one shot; most theory focuses on one-shot

procedures, but most experimental work has focused on dynamic

procedures

(b) What are the main issues and motivations?

i. experimental evidence dating from the 1960s (V. Smith, inspired

by E. Chamberlin’s classroom experiments at Harvard)

A. "clearinghouse" (one-shot) versus continuous time

ii. price discovery vs. price verification (R. Wilson, Reny and Perry

(2006)): where do prices come from?

iii. M. A. Satterthwaite: strategic behavior is a problem in small

markets but not in large markets, and it doesn’t require a lot of

traders for a market to be large

A. Myerson-Satterthwaite

B. Gibbard-Satterthwaite

iv. market design

A. design of algorithms for computerized trading

B. in parallel to the use of auction theory to inform the design

of auctions

C. Budish, Cramton and Shim (2014)

D. Loertscher and Mazzetti (2014), "A Prior-Free Approximately

Optimal Dominant-Strategy Double Auction"

2. Elementary model of a static double auction: independent, private values

(a) Chatterjee and Samuelson (1983) in the bilateral case

(b) buyers, each of whom wishes to buy at most one unit of an in-

divisible good, sellers, each of whom has one unit of the good to

sell

(c) redemption value/cost: ∈ [ ], v , ∈ [ ], v . Here,

for simplicity, [ ]=[ ] = [0 1]

(d) for ∈ [0 1], -double auction in bilateral case:i. bid , ask

ii. trade iff ≥ at price + (1− )

iii. = 1: buyer’s bid double auction

A. dominant strategy of seller to submit his cost as his ask

iv. ∈ (0 1)

1

v. FOCs, assuming increasing strategies:

( ) =

Z −1()

0

( − (+ (1− )()))()

( ) = ( − ¡+ (1− )(−1 ())¢(−1 ()) · 1

0 (−1 ())−

¡−1 ()

¢= ( − ) · (−1 ())

0 (−1 ())−

¡−1 ()

¢

( ) =

Z 1

−1()((() + (1− ))− )()

( ) = −(¡(−1()) + (1− )¢− )(−1())

· 1

0 (−1())+ (1− )

¡1−

¡−1 ()

¢¢= −(− ) · (−1())

0 (−1())+ (1− )

¡1−

¡−1 ()

¢¢equilibrium:

0 = ( −()) · (−1())0 (−1())

− ¡−1 (())

¢= −(()− ) · (−1(())

0 (−1(())+ (1− )

¡1−(−1()

¢vi. "linked" differential equations

vii. Chatterjee-Samuelson linear solution in uniform case:

() =

½23 + 1

12if ≥ 1

4

if ≥ 14

() =

½23+ 1

4if ≤ 3

4

if ≥ 34

viii. in general: every system of differential equations has a geometric

representation

0 ≤ ≤ ≤ 1

0 = ( − ) · ()̇− ()

0 = −(− ) · ()̇ + (1− ) (1− ())

ix. Increasing strategies:

A. probability of trading must be nondecreasing in a buyer’s

value and nonincreasing in a seller’s cost

B. "no flat spots" in the multilateral case

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x. Sufficiency of FOC: Sufficiency of FOC

evaluating the buyer’s FOC at bid , value −1():

0 = (−1 ()− ) · (−1()0 (−1())

− ¡−1 ()

¢⇔(−1 ()− ) =

¡−1 ()

¢0 (−1())

0¡−1()

¢marginal utility with value and bid :

( ) = ( − ) · (−1 ())0 (−1 ())

− ¡−1 ()

¢=

(−1 ())0 (−1 ())

"( − )−

¡−1 ()

¢(−1 ())

0¡−1 ()

¢#

=(−1 ())0 (−1 ())

£( − )− ¡−1()−

¢¤=

(−1 ())0 (−1 ())

£( −−1())

¤xi. existence of "double continuum" of equilibria

3

A. Leininger, Linhart and Radner (1989): step function equilib-

ria

B. Linhart and Radner (1989) : minmax regret and minmax

expected regret

(e) Multilateral case: bids, asks. Expressing market-clearing price

in terms of order statistics.

(1) ≤ (2) ≤ ≤ (+)

Assuming () (+1):

bids asks

≥ (+1)

≤ () −

4

Figure 1:

In the case of () = (+1): + 1

bids asks

() = (+1) − −

= () = (+1)

() = (+1) − −

We have + (− − ) ⇒ + ; there are enough units for

sale at = () = (+1) to satisfy every buyer who is willing to pay

more than this price. Similarly, + ( − − ) ⇒ + ,

and so there are enough buyers willing to buy at this price to satisfy

every seller who is willing to accept less than this price. At issue is

allocating among the + traders who bid/ask = () = (+1).

If + + , then there is excess demand at the price = () =

(+1); satisfy all of the buyers who bid more than the price and then

randomly allocate the remaining supply among those buyers whose

bids equaled the price. If + + , then we have excess supply

at the price. Allow all the sellers whose asks were below to sell and

then randomly choose among those whose asks equaled the price to

determine who gets to sell.

(f) no trade equilibrium

(g) the BBDA: = (+1) with sellers trading only if their asks are

strictly less than the price

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i. FOC:

( ) = ( − ) Pr( = ())− Pr(() (+1))

Pr(() (+1)) =

−1X=0

µ− 1

¶µ

−

¶¡−1()

¢·¡1−¡−1()

¢¢−1− ()

−(1− ())

−+

Pr( = ()) =

()·−1X=0

µ− 1

¶µ− 1

− 1−

¶¡−1()

¢ ¡1−

¡−1()

¢¢−1− · ()

−1−(1− ())

−+

+(−1) (−1())

0(−1())·−1X=0

µ− 2

¶µ− 2

− 1−

¶¡−1()

¢ ¡1−

¡−1()

¢¢−2− · ()

−1−(1− ())

−++1

ii. existence of equilibrium

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iii. uniform on [0 1]: ()= +1

iv. k-DA

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1. (a) Convergence results: , satisfy the bounds

1

≤

≤

8

for some constant . Let denote an equilibrium in the

market with buyers and sellers.

i. Convergence to price-taking behavior at the rate (1): There

exists a constant 1 () such that

−() ()− ≤ 1 ()

ii. Convergence to efficiency:

A. relative inefficiency:

−

B. relative inefficiency is (12), i.e., there exists a constant

2 () such that

−

≤ 2 ()

2

iii. meaningfulness of rates of convergence: statistics

A. numerical results

(b) Are these rates fast? M. A. Satterthwaite and S. R. Williams, "The

Optimality of a Simple Market Mechanism". Econometrica, Vol. 70,

No. 5 (Sep., 2002), pp. 1841-1863.

i. the k-DA is worst case asymptotic optimal among all Bayesian

incentive compatible, interim individually rational and ex ante

budget balanced mechanisms

A. asymptotic: mechanisms are compared using the rates at

which relative inefficiency converges to zero

B. worst-case: each mechanism is evaluated in its least favorable

environment (i.e., the choice of the distributions and )

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ii. Result

A. constrained efficient mechanism in the sense of Myerson

B. relative inefficiency of the constrained efficient mechanism is

at least 2 in the uniform case for some

C. We believe that this holds for all , that are reasonably

well-behaved

D. relative inefficiency of any mechanism in its worst case ≥relative inefficiency of constrained efficient mechanism in its

worst case ≥ 2

E. Applying a term from computer science, the main result of

this paper is that the k-DA is worst-case asymptotic optimal

among all mechanisms for organizing trade that satisfy these

two constraints. "Asymptotic" refers here to the ranking

of mechanisms using rates of convergence and "worst-case"

refers to the evaluation of each mechanism in its least favor-

able environment for each value of m. Stated simply, this

result means that the k-DA’s worst-case error over a set of

environments converges to zero at the fastest possible rate

among all interim individually rational and ex ante budget

balanced mechanisms.

2. Related Work

(a) T. A. Gresik and M. A. Satterthwaite, "The Rate at Which a Simple

Market Converges to Efficiency as the Number of Traders Increases:

An Asymptotic Result for Optimal Trading Mechanisms", J. of Econ.

Theory 48 (1989), pp. 304-332.

i. Along with Wilson (1985), these two papers are to first to go be-

yond the bilateral bargaining model of Chatterjee and Samuelson

(1983) to the multilateral case.

ii. markets with buyers and sellers for ∈ Niii. relative inefficiency in the constrained efficient mechanism is

³ln 2

´,

a result that has been superceded by the rate established for the

-DA

iv. notable for the question that it pursues, which is the origin of all

of the convergence results that I discuss above

(b) R. Wilson, "Incentive Efficiency of Double Auctions". Econometrica,

Vol. 53, No. 5 (Sep., 1985), pp. 1101-1115.

i. If min {} is sufficiently large, then an equilibrium of a -

double auction is interim incentive efficient in the sense of Holm-

ström and Myerson (1983)

A. It cannot be common knowledge at the interim stage that

some other equilibrium of some possibly different procedure

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Pareto dominates the equilibrium of the -double auction

under consideration

B. endurance of simple procedures such as the -double auction

ii. assumes existence of increasing, differentiable equilibrium strate-

gies with derivatives bounded uniformly for all ,

iii. imputes welfare weights from the allocation rule in the -DA in

equilibrium

A. the issue is whether or not these imputed weights are positive

as required for interim incentive efficiency

B. these imputed weights converge uniformly for all trader types

to 1 as min {} → ∞, and so for large min {} theyare positive for all trader types

C. no intuition as to why a large market is required for interim

incentive efficiency; market size is a means to an end

D. no examples of small markets in which an equilibrium fails to

be interim incentive efficient, except the no-trade equilibrium

iv. two aspects of the Wilson Critique:

A. procedures that are not defined in terms of the probabilistic

beliefs of the agents (traders)

B. relaxing the assumption of common knowledge of beliefs

C. p. 1114: "The practical advantage of a double auction is

that its rules for trades and payments do not invoke the data

that are common knowledge among the agents — namely, the

numbers of buyers and sellers, the joint probability distrib-

ution of their types, and the functional dependence of their

reservation prices on the type parameter. Instead, the bur-

den of coping with the complexity of the common knowledge

features is assumed by the traders in the construction of their

strategies."

(c) P. McAfee, "A Dominant Strategy Double Auction." Journal of Eco-

nomic Theory, Vol. 56, No. 2 (April 1992), pp. 434-450.

i. When ((+1) + (+1))2 ∈ [() ()], = ((+1) + (+1))2

and trades are made.

ii. When ((+1) + (+1))2 ∈ [() ()], highest − 1 buyers pay(), lowest − 1 sellers receive (), − 1 trades made andmonetary surplus of ( − 1)(() − ())

iii.

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iv. virtues:

A. dominant strategies

B. if the monetary surplus of the "specialist" is counted among

the gains from trade, then expected inefficiency is (1 (+ ))

v. flaw: does not converge to efficiency if the monetary surplus is

treated as a cost of trading to the traders

vi. Loertscher and Marx (2015)

(d) Large Double Auctions

i. approach of my work with Satterthwaite

A. focus on the first order conditions

B. analyzed using combinatorics

ii. large double auctions: assumes a sufficiently large number of

traders

A. results of probability and statistics become applicable

B. asymptotics

C. remains a model of strategic price discovery

D. rarely the production of an equilibrium or any connection to

smaller markets

E. motivation typically based upon longstanding problems in

microeconomic theory: strategic foundation for competitive

equilibrium and for REE

12

iii. M. W. Cripps and J. M. Swinkels, "Efficiency of Large Double

Auctions". Econometrica, Vol. 74, No. 1 (Jan., 2006), pp. 47-

92.

A. result: As the number fo traders grows every nontrivial equi-

librium of the double auction setting converges to the Wal-

rasian outcome. Relative inefficiency disappears at the rate

12− for any 0

B. two goods, initially unitary supply/demand, later multiple

units are allowed

C. considers sequences of markets with possibly correlated, pri-

vate values in [0 1]

D. ≤symmetry of the distribution and across the strategies usedby each side of the market is allowed

E. Cripps and Swinkels: symmetry of strategies assumes away

the problem of making sure that the units are allocated prop-

erly to each side of the market

F. restriction on distribution: no asymptotic gaps, no asymp-

totic atoms

G. for ∈ (0 1], z-independence: bounds the amount that thedistribution of any trader’s value changes conditional on the

values of the remaining traders

H. asymmetry and "purification" of equilibrium strategies as the

number of traders grows

I. to establish the rate of convergence, "quite large" is neces-

sary. No indication of when the rate begins to be observed.

iv. P. J. Reny and M. Perry, "Toward a Strategic Foundation for

Rational Expectations Equilibrium", Econometrica, Vol. 74, No.

5 (Sep., 2006), pp. 1231-1269.

A. provide a strategic foundation for rational expectations equi-

librium

B. affiliated, interdependent values/costs

C. limit market, with a continuum of traders and real-valued

bids/asks: Bayes-Nash equilibrium in increasing strategies

that implements a fully-revealing REE price

D. main result is continuity as the number of traders and the

number of possible bids/asks goes to infinity: for a suffi-

ciently large number of traders, and for a discrete grid of

possible bids/asks that is sufficiently fine, there exists a BNE

in increasing strategies that approximates the equilibrium of

the limit market

E. all traders are fully rational and strategic: no noise traders

and sellers are active (unlike auction models)

F. No indication of how large a market is required, no examples

in finite markets

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G. "Our main insight is that establishing the existence of a

monotone equilibrium poses serious difficulties only when in-

dividual agents can have a significant impact on the price."

Is this a problem of "proof", or truly of existence? They

suggest the later.

(e) R. C. Shafer, "Convergence to Price-Taking by Regret-Minimizers in

-Double Auctions."

i. P. B. Linhart and R. Radner, "Minimax-Regret Strategies for

Bargaining over Several Variables". J. of Econ. Theory 48, 152-

178 (1989).

ii. Shafer: Alternatives to Bayesian decision-making in modeling

how traders select their bids and asks in a -DA

iii. Does the emergence of price-taking behavior as the market in-

creases in size fundamentally require that one be a Bayesian?

iv. minimax regret and maxmin: behavior invariant to the size of

the market

v. culprit: this is true of any decision rule that satisfies the axiom

of symmetry

vi. Γ-minimax regret, and Γ-maxmin; minimizing maximum expected

regret

(f) J. H. Kagel andW. Vogt, "Buyer’s Bid Double Auctions: Preliminary

Experimental Results". Chapter 10 in The Double Auction Market:

Institutions, Theories and Evidence, D. Friedman and J. Rust, eds.

Addison Wesley (1993): Reading.

i. experimental design

A. = 2 and = 8 traders on each side

ii. few sellers played their dominant strategies, causing inefficiency

iii. buyers underbid by less than the equilibrium prediction

iv. change from = 2 to = 8 notable but not as much as pre-

dicted by theory

A. this is largely attributable to the fact that the underbidding

by buyers in the case of = 2 is not as extreme as theory

predicts, leaving little room for improvement

v. opportunities for learning in BBDA

(g) Continuous Bid/Ask Market

i. R. Wilson, "Equilibria of Bid-Ask Markets," in: Arrow and the

Ascent of Economic Theory: Essays in Honor of Kenneth J.

Arrow, G. Feiwel (ed.); Chapter 11, pp. 375-414. London and

New York: Macmillan Press and New York University Press,

1987.

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ii. D. Easley and J. Ledyard, "Theories of Price Formation and

Exchange in Double Oral Auctions," in: The Double Auction

Market: Institutions, Theories and Evidence, D. Friedman and

J. Rust, eds. Addison Wesley (1993): Reading.

iii. D. Friedman, "A Simple Testable Model of Double Auction Mar-

kets". J. Econ. Behav. & Org. (1991), 47-70.

iv. D. Friedman, "The Double Auction Market Institution: A Sur-

vey", in: The Double Auction Market: Institutions, Theories

and Evidence, D. Friedman and J. Rust, eds. Addison Wesley

(1993): Reading.

3. Asymptotics

(a) goals

i. identify the asymptotic distribution of the BBDA’s price

ii. identify the asymptotic limits of the probabilities in a trader’s

FOC

iii. formulate the asymptotic FOCs (AFOCs) and solve

iv. compare the solutions to the AFOCs to computed equilibrium

v. AFOCs identify what is "first order" in a trader’s decision prob-

lem

A. comparative statics in the distributions and the numbers of

traders

(b) review

i. CPV environment

ii. informational model

iii. convergence results

iv. limit market: Let ≡ (+) and (ksi) ≡ −1 (), the th

quantile of distribution . The limit market in state consists

of probability masses of measure of buyers and measure 1−

of sellers with values/costs , which conditional on are i.i.d.

according to ( − ).

v. REE: The unique REE price in the limit market in state is

REE ≡ +

(c) fix , ; markets with buyers and sellers

i. () = :(+)−1, () = +1:(+)−1, () = +1:(+)

ii. (), () and e() are asymptotically consistent, unbiased and

normal estimators of the REE price in state :

() () ∼ ANµREE

(+ )2

[(+ )− 1] 2()¶

e() ∼ ANµREE

(+ )2

(+ )2()

¶

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iii. figures that depict the distribution of e()

(d) FOC:

( − )()|(|)− Pr[() ≤ ≤ ()|] = 0i. asymptotic offset:

approx () =1

(+ ) − 11

()

A. prices centered on REE ≡ + B. no distinction between and except in determining

C. dependence on ()

(e) Numerical Example

i.

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A. note: comparison between table

B. accuracy of approximation

ii.

17